This is the first of two activities in which students define sequences with equations and use their equations to answer questions about the context (MP2). The population values were purposefully chosen in order for students to focus on creating representations (like tables and graphs) and not on calculating “best fit.”

Making spreadsheet and graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

If students identify the arithmetic sequence as geometric or vice versa, consider saying:

• “Explain your representation to me.”
• “How could making a table of values for your equation help you check your work?”

The goal of this discussion is for students to understand how different representations of functions are useful in different ways. For example, using technology to make a graph from the definition of the term of each sequence can efficiently lead to the solution while testing values or trying to write an equation to solve may be more time consuming or challenging.

Select students to share what kind of sequence is represented by Population A and their recursive or term definitions. Then do the same for Population B. Record for all to see any representations, such as extra rows in the table, graphs, or spreadsheets, that students made to write their equations. If not brought up in student explanations, ask how each representation shows different features of the sequence, for example, the growth factor or rate of change.

Conclude the discussion by inviting students to share how they determined whether Population B would ever overtake Population A, highlighting any student-created representations used to answer this question.

This is the second of two activities in which students define sequences with equations and use their equations to answer questions about the context. Similar to the previous activity, students are asked to "write an equation" without being told the type, so the choice is theirs to define the sequences recursively or for the term, after they make sense of the problem (MP1). Making spreadsheet and graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Monitor for students who use different strategies to identify the type of sequences represented by the black and white squares. Here are some examples of strategies students may use, ordered from concrete to abstract:

• Graphing the number of white or black squares for each step
• Making a table for the total number of white or black squares for each step and then calculating the differences between successive terms
• Visualizing ways that the white squares grow to write equations in order to tell that is arithmetic
• Visualizing ways that the black squares grow to write equations in order to tell that is neither arithmetic nor geometric

If a strategy is not used, it is not necessary to introduce it.

The routine of Anticipate, Monitor, Select, Sequence, Connect (5 Practices) requires a balance of planning and flexibility. The anticipated approaches might not surface in every class, and there may be reason to change the order in which strategies are presented. While monitoring, keep in mind the learning goal and adjust the order to ensure all students have access to the first idea presented (whether that be a common misconception or a different approach).

If students are not sure how to identify what types of sequences and are, consider asking:

• “How did you decide what types of sequences the populations were in the earlier activity?”
• “How could you use a table to help you decide?”

The purpose of this discussion is to share different ways of using equations to define sequences and highlighting some of the advantages and disadvantages of different methods for identifying what model to use.

Invite previously selected students to share how they determined if and were arithmetic, geometric, or neither. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

Connect the different responses to the learning goals by asking questions, such as:

• “A graph of ’s information made it clear that the function is arithmetic. What could we tell from a graph of ’s values?” (The graph curved, so we knew wasn’t arithmetic, but we couldn’t tell from the graph if was geometric or not.)
• “What advantages and disadvantages did tables have for identifying the type of function?” (Tables help us efficiently determine the rate of change or growth factor between terms, but more work is needed to figure out how to write an equation for a function that is neither arithmetic nor geometric.)
• “How did visualizing the way the black squares grow help write an equation for ?” (By thinking of the black squares as a rectangle that grows from 1 by 2 to 2 by 3 to 3 by 4, it was clear that, for Step , the rectangle would be .)